Relativity and cosmology

Special Relativity Theory And Black Holes

It is possible to find the relationship between the mass and the radius of a spherical black hole keeping in mind that the maximum speed that could reach an object, according to the theory of relativity, it is the speed of light.

The scape velocity in the surface of a spherical star will be the maximum velocity that could reach an object in order to stay in orbit around the star. This will happen when the kinetic energy of one object is equal to the potential energy due to the star gravitational attraction.

The kinetic energy (Ec) according to the classical mechanics is

Ec=½ mv² (1)

and the potential energy is Ep=GmM/r (2)

being v the speed of the object in orbit, m the mass of the object in orbit, M the mass of the star, r the distance from the center of the star until the point where it is the object in orbit and G the universal gravitation constant.

Equaling the potential energy with the kinetic energy and isolating the velocity we obtain the equation of the speed of escape:

(3)

then for a scape velocity equal to the speed of the c light and isolating M/r in the previous formula we can obtain

(4)

as c=2.99793 x 108 m/s and G=6.6732 x 10-11 Nm²/kg² we obtain

M/r=6.734 x 1026 kg/m

this is the relationship between the mass and the radius of a spherical body in order to be a black hole. With this relationship we could find the radius that must have diverse stellar objects in order to be a black hole.